| L(s) = 1 | + 3·3-s − 0.981·5-s − 17.7·7-s + 9·9-s − 61.8·13-s − 2.94·15-s + 130.·17-s + 116.·19-s − 53.3·21-s − 26.4·23-s − 124.·25-s + 27·27-s + 179.·29-s − 319.·31-s + 17.4·35-s − 177.·37-s − 185.·39-s − 338.·41-s + 159.·43-s − 8.83·45-s + 98.9·47-s − 27.0·49-s + 390.·51-s + 562.·53-s + 348.·57-s + 443.·59-s + 551.·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.0877·5-s − 0.959·7-s + 0.333·9-s − 1.32·13-s − 0.0506·15-s + 1.85·17-s + 1.40·19-s − 0.554·21-s − 0.240·23-s − 0.992·25-s + 0.192·27-s + 1.15·29-s − 1.85·31-s + 0.0842·35-s − 0.790·37-s − 0.762·39-s − 1.29·41-s + 0.566·43-s − 0.0292·45-s + 0.307·47-s − 0.0789·49-s + 1.07·51-s + 1.45·53-s + 0.810·57-s + 0.979·59-s + 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.115311680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.115311680\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 0.981T + 125T^{2} \) |
| 7 | \( 1 + 17.7T + 343T^{2} \) |
| 13 | \( 1 + 61.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 130.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 26.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 179.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 319.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 338.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 159.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 98.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 562.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 443.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 551.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 454.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 214.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 162.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 167.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 357.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.11e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.348360309630454486367336628361, −8.301874279329914691545308618291, −7.44037443277762354661231702593, −7.03090073487094785333436287831, −5.73404236253699692137370626028, −5.09889974174098817647352097369, −3.68854367590338346531325637552, −3.20309187692979376879693112211, −2.08651450210452763893159733516, −0.68337440302937781824303988550,
0.68337440302937781824303988550, 2.08651450210452763893159733516, 3.20309187692979376879693112211, 3.68854367590338346531325637552, 5.09889974174098817647352097369, 5.73404236253699692137370626028, 7.03090073487094785333436287831, 7.44037443277762354661231702593, 8.301874279329914691545308618291, 9.348360309630454486367336628361
